Consider two containers A and B containing identical gases at the same pressure, volume and temperature. The gas in container A is compressed to half of its original volume isothermally while the gas in container B is compressed to half of its original value adiabatically. The ratio of final pressure of gas in B to that of gas in A is

Options

A. `2^(gamma-1)`
B. `(1/2)^(gamma-1)`
C. `(1/(1-gamma))^(2)`
D. `(1/(gamma-1))^(2)`

Explanation

To solve the problem, we will analyze the two processes: isothermal compression in container A and adiabatic compression in container B.

### Step 1: Analyze the isothermal process in container A
For an isothermal process, the relationship between pressure and volume is given by Boyle's Law, which states that \( P_1 V_1 = P_2 V_2 \) (where \( P_1 \) and \( V_1 \) are the initial pressure and volume, and \( P_2 \) and \( V_2 \) are the final pressure and volume).

Given that the gas in container A is compressed to half of its original volume:
- \( V_f = \frac{1}{2} V_i \)

Using Boyle's Law:
\[ P_f = P_i \frac{V_i}{V_f} = P_i \frac{V_i}{\frac{1}{2} V_i} = P_i \cdot 2 = 2 P_i \]

### Step 2: Analyze the adiabatic process in container B
For an adiabatic process, the relationship is given by:
\[ P_1 V_1^\gamma = P_2 V_2^\gamma \]
where \( \gamma \) is the heat capacity ratio (specific heat at constant pressure divided by specific heat at constant volume).

Again, since the gas in container B is also compressed to half of its original volume:
- \( V_f = \frac{1}{2} V_i \)

Using the adiabatic condition:
\[ P_f = P_i \left( \frac{V_i}{V_f} \right)^\gamma = P_i \left( \frac{V_i}{\frac{1}{2} V_i} \right)^\gamma = P_i \cdot 2^\gamma \]

### Step 3: Calculate the ratio of final pressures
Now we need to find the ratio of the final pressure in container B to the final pressure in container A:
\[ \frac{P_{fB}}{P_{fA}} = \frac{P_i \cdot 2^\gamma}{2 P_i} \]

The \( P_i \) cancels out:
\[ \frac{P_{fB}}{P_{fA}} = \frac{2^\gamma}{2} = 2^{\gamma - 1} \]

### Conclusion
Thus, the ratio of the final pressure of gas in container B to that of gas in container A is:
\[ \frac{P_{fB}}{P_{fA}} = 2^{\gamma - 1} \]